Solve (frac{r^{2}s^{-4}t^{6}}{r^-4s^2t^6})^1/6

$$( \frac { r ^ { 2 } s ^ { - 4 } t ^ { 6 } } { r ^ { - 4 } s ^ { 2 } t ^ { 6 } } ) ^ { 1 / 6 }$$

Evaluate

$$\frac{|r|}{|s|}$$

Solution Steps

Cancel out t^{6} in both numerator and denominator.

$$\left(\frac{s^{-4}r^{2}}{r^{-4}s^{2}}\right)^{\frac{1}{6}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\left(\frac{s^{-4}r^{6}}{s^{2}}\right)^{\frac{1}{6}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\left(\frac{r^{6}}{s^{6}}\right)^{\frac{1}{6}}$$

To raise \frac{r^{6}}{s^{6}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(r^{6}\right)^{\frac{1}{6}}}{\left(s^{6}\right)^{\frac{1}{6}}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and \frac{1}{6} to get 1.

$$\frac{r^{1}}{\left(s^{6}\right)^{\frac{1}{6}}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and \frac{1}{6} to get 1.

$$\frac{r^{1}}{s^{1}}$$

Calculate r to the power of 1 and get r.

$$\frac{r}{s^{1}}$$

Calculate s to the power of 1 and get s.

$$\frac{r}{s}$$

Differentiate w.r.t. s

$$-\frac{|r||s|}{s^{3}}$$